1969 : A one-dimensional noninvertible map

Before the emergence of chaos in electronic circuits during the late 70s and early 80s there is an important contribution by Igor Gumowski & Christian Mira (the “Toulouse Research Group”) whose a short story was given in Ref. [1] Stimulated by a paper by C. P. Pulkin [2] who showed that in one-dimensional noninvertible map infinitely many unstable cycles may lead to bounded complex iterated sequences, Gumowski & Mira studied the piecewise-linear map [3]

For some -values, they obtained attractive limit set made of bounded cloud of points as shown in Fig. 1 (). This could have been the first now called “chaotic” solution to a piecewise-linear map reported with an explicit picture. By these times, Gumowski & Mira indicated these types of behaviors as “Pulkin phenonmenon”. This is only ten years later that first chaotic solutions were widely investigated in
electronic circuits.

Fig. 1. Bounded-2 chaotic attractor solution to the piecewise-linear map as published in 1969

Increasing slightly the value to 2.39, the chaotic solution observed is just before a boundary crisis unifying the two sets of points (Fig. 2).

Fig. 2. Chaotic solution just before a boundary crisis.

Budapest, 1968

[1] C. Mira, I. Gumowski and a Toulouse Research Group in the “prehistoric times of chaotic dynamics”, World Scientific Series in Nonlinear Science A, 39, 95-198, 2000.